On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system

Understand

We suggest a Hamiltonian formulation for the spin Ruijsenaars-Schneider system in the trigonometric case.

• Within this interpretation, the phase space is obtained by a quasi-Hamiltonian reduction performed on (the cotangent bundle to) a representation space of a framed Jordan quiver.
• For arbitrary quivers, analogous varieties were introduced by Crawley-Boevey and Shaw, and their interpretation as quasi-Hamiltonian quotients was given by Van den Bergh.
• Using Van den Bergh's formalism, we construct commuting Hamiltonian functions on the phase space and identify one of the flows with the spin Ruijsenaars-Schneider system.